Finite Group Databasebeta

← C7×#18929 #13196145

Subgroups — C7×#18929 #13196145

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3506C211yesmaximal, characteristic
2C7×#3230C211yesmaximal, characteristic, complemented
3C7×#3206C211yesmaximal, characteristic, complemented
4C2×C7×#474C211yesmaximal, characteristic, complemented
5C7×#1647C214yesmaximal, complemented
6C7×#1646C214yesmaximal, complemented
7C7×#1542C211yesmaximal, characteristic, complemented
8C7×#1480C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic, complemented
10C7×#559—22no—
11C7×#556—21no—
12C7×#555C2212yes—
13C7×#553—22no—
14C7×#550—21no—
15C7×#547C2212yes—
16C7×#538—22no—
17C7×#535—21no—
18C7×#533C2211yescharacteristic
19C7×#532C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C7×#525—22no—
22C7×#520—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C7×#500C2212yescomplemented
25C7×#492—21no—
26C7×#487—21no—
27C7×#487C2212yescomplemented
28C7×#484—22no—
29C7×#477—22no—
30C7×#474—22no—
31C7×#474C2214yes—
32C7×#461C2214yescomplemented
33C7×#448—22no—
34C7×#431C2211yescharacteristic
35C7×#431C2211yescharacteristic
36C2×C7×#107—21no—
37C2×C7×C4⋊C8C2211yescharacteristic
38C2×C7×#102—21no—
39C7×#333C2212yescomplemented
40C7×#332C2212yescomplemented
41C7×D4⋊C8C2214yes—
42C7×D4⋊C8C2214yescomplemented
43#18929C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—22no—
45C2×C7×#41—21no—
46C22×C7×Q8—21no—
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—21no—
49C2×C7×SD16—24no—
50C22×C7×C8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8D411yescharacteristic
55C7×C4⋊Q8C2312yes—
56C7×C4⋊Q8C2311yescharacteristic
57C7×C4⋊Q8C2311yescharacteristic
58C7×#127—22no—
59C7×#127D411yescharacteristic
60C7×#126—24no—
61C7×#125—21no—
62C7×#124—22no—
63C7×#124—24no—
64C7×#124—21no—
65C7×#123—24no—
66C7×#123—24no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8D412yes—
70C7×#121—24no—
71C7×#121—22no—
72C7×#121—22no—
73C7×#121—22no—
74C7×#121—21no—
75C7×#121D412yes—
76C7×#120—24no—
77C4×C7×Q8—22no—
78C4×C7×Q8—21no—
79C4×C7×Q8D412yes—
80C4×C7×D4—24no—
81C4×C7×D4—22no—
82C4×C7×D4—24no—
83C4×C7×D4—21no—
84C4×C7×D4—21no—
85C4×C7×D4—21no—
86C4×C7×D4D412yes—
87C4×C7×D4C2312yes—
88C4×C7×D4C2312yes—
89C7×#117—21no—
90C7×#117—22no—
91C7×#117—22no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C22⋊C4—21no—
97C2×C42×C7C2311yescharacteristic
98C7×#107—24no—
99C7×#107—24no—
100C7×C4⋊C8—21no—
101C7×C4⋊C8—21no—
102C7×C4⋊C8C2312yes—
103C7×C4⋊C8C2312yes—
104C7×#103—24no—
105C7×#103—24no—
106C7×#102—24no—
107C7×C22⋊C8—21no—
108C7×C22⋊C8—21no—
109C4×C7×C8—21no—
110C4×C7×C8C2312yes—
111#3506C2×C711yescharacteristic
112#3230C2×C711yescharacteristic, complemented
113#3206C2×C711yescharacteristic, complemented
114C2×#474C2×C711yescharacteristic, complemented
115#1647C2×C714yescomplemented
116#1646C2×C714yescomplemented
117#1542C2×C711yescharacteristic, complemented
118#1480C2×C711yescharacteristic, complemented
119#1478C2×C711yescharacteristic, complemented
120C24×C7—21no—
121C7×#41—42no—
122C7×#41—42no—
123C7×#41—84no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—41no—
128C2×C7×Q8—82no—
129C2×C7×Q8C2×D412yes—
130C2×C7×D4—22no—
131C2×C7×D4—24no—
132C2×C7×D4—24no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4C2×D412yes—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—24no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—41no—
146C22×C4×C7—82no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C7×SD16—84no—
151C2×C7×C8—22no—
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—42no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4—41no—
167C7×C4⋊C4C2×D412yes—
168C7×C4⋊C4C2×D412yes—
169C7×C4⋊C4C2×D412yes—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—24no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C42×C7—24no—
179C42×C7—22no—
180C42×C7C2×D411yescharacteristic
181C42×C7C2×D411yescharacteristic
182C42×C7C2×D411yescharacteristic
183C42×C7C2411yescharacteristic
184#559—22no—
185#556—21no—
186#555C22×C712yes—
187#553—22no—
188#550—21no—
189#547C22×C712yes—
190#538—22no—
191#535—21no—
192#533C22×C711yescharacteristic
193#532C22×C711yescharacteristic
194C2×C4⋊Q8C22×C711yescharacteristic
195#525—22no—
196#520—21no—
197C2×C4×D4C22×C711yescharacteristic
198#500C22×C712yescomplemented
199#492—21no—
200#487—21no—
201#487C22×C712yescomplemented
202#484—22no—
203#477—22no—
204#474—22no—
205#474C22×C714yes—
206#461C22×C714yescomplemented
207#448—22no—
208#431C22×C711yescharacteristic
209#431C22×C711yescharacteristic
210C2×#107—21no—
211C2×C4⋊C8C22×C711yescharacteristic
212C2×#102—21no—
213#333C22×C712yescomplemented
214#332C22×C712yescomplemented
215D4⋊C8C22×C714yes—
216D4⋊C8C22×C714yescomplemented
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—42no—
220C23×C7—81no—
221C23×C7—81no—
222C23×C7#12011yescharacteristic
223C7×Q8—42no—
224C7×Q8—42no—
225C7×Q8—42no—
226C7×Q8—44no—
227C7×Q8—82no—
228C7×D4—22no—
229C7×D4—22no—
230C7×D4—24no—
231C7×D4—44no—
232C7×D4—42no—
233C7×D4—42no—
234C7×D4—81no—
235C7×D4—84no—
236C7×D4—84no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—41no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—82no—
248C2×C4×C7—81no—
249C2×C4×C7—84no—
250C2×C4×C7—82no—
251C2×C4×C7#12011yescharacteristic
252C2×C4×C7#12011yescharacteristic
253C2×C4×C7#12011yescharacteristic
254C2×C4×C7#13512yes—
255C2×C4×C7#13712yes—
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C7×C8—42no—
260C7×C8—42no—
261C7×C8—42no—
262C2×#41—22no—
263C2×#41—21no—
264C22×Q8—21no—
265C22×D4C7×D411yescharacteristic
266C23×C4—21no—
267C2×SD16—24no—
268C22×C8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8C7×D411yescharacteristic
273C4⋊Q8C23×C712yes—
274C4⋊Q8C23×C711yescharacteristic
275C4⋊Q8C23×C711yescharacteristic
276#127—22no—
277#127C7×D411yescharacteristic
278#126—24no—
279#125—21no—
280#124—22no—
281#124—24no—
282#124—21no—
283#123—24no—
284#123—24no—
285C22⋊Q8—21no—
286C22⋊Q8—22no—
287C22⋊Q8C7×D412yes—
288#121—24no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—21no—
293#121C7×D412yes—
294#120—24no—
295C4×Q8—22no—
296C4×Q8—21no—
297C4×Q8C7×D412yes—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—24no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C7×D412yes—
305C4×D4C23×C712yes—
306C4×D4C23×C712yes—
307#117—21no—
308#117—22no—
309#117—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C7×D411yescharacteristic
314C2×C22⋊C4—21no—
315C2×C42C23×C711yescharacteristic
316#107—24no—
317#107—24no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8C23×C712yes—
321C4⋊C8C23×C712yes—
322#103—24no—
323#103—24no—
324#102—24no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327C4×C8—21no—
328C4×C8C23×C712yes—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7—84no—
338C22×C7#45212yes—
339C22×C7C2×#12011yescharacteristic
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—21no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—82no—
348C4×C7#45212yes—
349C4×C7C2×#13511yescharacteristic
350C4×C7C2×#13711yescharacteristic
351C4×C7#57612yes—
352C24—21no—
353#41—42no—
354#41—42no—
355#41—84no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—82no—
361C2×Q8C2×C7×D412yes—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4C2×C7×D412yes—
371C2×D4C2×C7×D412yes—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—82no—
379C22×C4C2×C7×D411yescharacteristic
380C22×C4C2×C7×D411yescharacteristic
381C22×C4C2×C7×D411yescharacteristic
382SD16—84no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—42no—
387C4⋊C4—24no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4C2×C7×D412yes—
400C4⋊C4C2×C7×D412yes—
401C4⋊C4C2×C7×D412yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C42—24no—
411C42—22no—
412C42C2×C7×D411yescharacteristic
413C42C2×C7×D411yescharacteristic
414C42C2×C7×D411yescharacteristic
415C42C24×C711yescharacteristic
416C2×C7—22no—
417C2×C7—82no—
418C2×C7—82no—
419C2×C7C2×#45211yescharacteristic
420C2×C7#300811yescharacteristic
421C2×C7#302211yescharacteristic
422C23—22no—
423C23—22no—
424C23—42no—
425C23—81no—
426C23—81no—
427C23C7×#12011yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—44no—
432Q8—82no—
433D4—22no—
434D4—22no—
435D4—24no—
436D4—44no—
437D4—42no—
438D4—42no—
439D4—81no—
440D4—84no—
441D4—84no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4C7×#12011yescharacteristic
457C2×C4C7×#12011yescharacteristic
458C2×C4C7×#12011yescharacteristic
459C2×C4C7×#13512yes—
460C2×C4C7×#13712yes—
461C2×C4C22×C7×D411yescharacteristic
462C2×C4C22×C7×D411yescharacteristic
463C2×C4C22×C7×D411yescharacteristic
464C8—42no—
465C8—42no—
466C8—42no—
467C7#1892911yescharacteristic, direct product factor, Sylow
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—84no—
477C22C7×#45212yes—
478C22C2×C7×#12011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4C7×#45212yes—
488C4C2×C7×#13511yescharacteristic
489C4C2×C7×#13711yescharacteristic
490C4C7×#57612yes—
491C2—22no—
492C2—82no—
493C2—82no—
494C2C2×C7×#45211yescharacteristic
495C2C7×#300811yescharacteristic
496C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3506C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#3230C2semidirect
3. C7×#3206C2semidirect
4. C2×C7×#474C2semidirect
5. C7×#1647C2semidirect
6. C7×#1646C2semidirect
7. C7×#1542C2semidirect
8. C7×#1480C2semidirect
9. C7×#1478C2semidirect
24. C7×#500C22semidirect
27. C7×#487C22semidirect
32. C7×#461C22semidirect
39. C7×#333C22semidirect
40. C7×#332C22semidirect
42. C7×D4⋊C8C22semidirect
43. #18929C7direct (N × H)
112. #3230C2×C7semidirect
113. #3206C2×C7semidirect
114. C2×#474C2×C7semidirect
115. #1647C2×C7semidirect
116. #1646C2×C7semidirect
117. #1542C2×C7semidirect
118. #1480C2×C7semidirect
119. #1478C2×C7semidirect
198. #500C22×C7semidirect
201. #487C22×C7semidirect
206. #461C22×C7semidirect
213. #333C22×C7semidirect
214. #332C22×C7semidirect
216. D4⋊C8C22×C7semidirect
467. C7#18929direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#35062
C7×#32302
C7×#32062
C2×C7×#4742
C7×#16472
C7×#16462
C7×#15422
C7×#14802
C7×#14782
#189297